The line segment shown below connects the points (1, 2) and (3, –2). As we look at this diagram from left to right, the segment slopes downward. We will define a negative number to describe the downward slope and how steeply this line segment slopes. To do this, we notice that the vertical distance (rise) between these points is 4 and the horizontal distance (run) is 2. The slope is a ratio of vertical to horizontal distances: .

Shown below is another segment of the same lengthperpendicular at (1, 2). The other endpoint is (– 3, 0). The rise between endpoints is 2 and the run is 4, the opposite of the rise and run for the segment connecting (1, 2) and (3, – 2). Also this new segment is sloping upward. The slope will be positive. We have slope = . Contrast this with the slope of the perpendicular which is – 2. This is the reciprocal, and differs in sign.

Summary:

If two lines are perpendicular, their slopes are negative reciprocals.

An extension of this example would be to calculate the equation of the line that is perpendicular to our original line, and passes through the same y-intercept, (0,7). Using the slope-intercept form of a line, y = mx + b, we can generate its equation as y = x + 7. As shown below, these two lines would cross on the y-axis and be oriented at 90º to each other. Note that the dimensions of the graphing calculator's screen unfortunately do not give the correct "impression of perpendicularity."